| L(s) = 1 | + (1.01 + 1.16i)2-s + (0.841 + 0.540i)3-s + (−0.0556 + 0.386i)4-s + (0.996 − 2.18i)5-s + (0.220 + 1.53i)6-s + (−0.959 − 0.281i)7-s + (2.09 − 1.34i)8-s + (0.415 + 0.909i)9-s + (3.55 − 1.04i)10-s + (1.41 − 1.63i)11-s + (−0.255 + 0.295i)12-s + (−1.10 + 0.325i)13-s + (−0.642 − 1.40i)14-s + (2.01 − 1.29i)15-s + (4.44 + 1.30i)16-s + (0.142 + 0.989i)17-s + ⋯ |
| L(s) = 1 | + (0.715 + 0.826i)2-s + (0.485 + 0.312i)3-s + (−0.0278 + 0.193i)4-s + (0.445 − 0.975i)5-s + (0.0898 + 0.624i)6-s + (−0.362 − 0.106i)7-s + (0.740 − 0.475i)8-s + (0.138 + 0.303i)9-s + (1.12 − 0.330i)10-s + (0.426 − 0.492i)11-s + (−0.0738 + 0.0852i)12-s + (−0.307 + 0.0902i)13-s + (−0.171 − 0.375i)14-s + (0.520 − 0.334i)15-s + (1.11 + 0.326i)16-s + (0.0345 + 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.51772 + 0.580750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51772 + 0.580750i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (1.93 - 4.38i)T \) |
| good | 2 | \( 1 + (-1.01 - 1.16i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (-0.996 + 2.18i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-1.41 + 1.63i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.10 - 0.325i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.142 - 0.989i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.102 - 0.710i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.0170 + 0.118i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.68 - 1.08i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.66 + 5.83i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.45 - 9.75i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.290 + 0.186i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 0.954T + 47T^{2} \) |
| 53 | \( 1 + (0.882 + 0.259i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (11.4 - 3.36i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (4.09 - 2.63i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.328 - 0.379i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-9.83 - 11.3i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.43 + 10.0i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.41 + 2.17i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.78 - 3.90i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (7.66 + 4.92i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.71 + 5.94i)T + (-63.5 - 73.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96573574103580203119662147395, −9.897300661844782089366940387018, −9.274022590667835694081879002524, −8.286596424503824314297245503018, −7.28576929793472549682852723948, −6.19522607244283841297784709997, −5.39526005512199650871971277032, −4.50067204079671253363343057289, −3.46697867837749218434076412556, −1.53026988889640059793478445799,
1.98200765851429422290050989654, 2.81701350048230450154673216886, 3.74455682393858529507111710255, 4.95348966351237669050221733494, 6.37527638061174162779618217769, 7.12870610771361819988029655740, 8.159421578644770058348446193006, 9.365920286044047400433324737867, 10.26545354341830292484293763269, 10.93160485607764505840935652927
